For $Y,Y^+$ three-dimensional smooth varieties related by a flop, Bondal and Orlov conjectured that the derived categories $D^b({\rm coh}(Y))$ and $D^b({\rm coh}(Y^+))$ are equivalent. This conjecture was recently proved by Bridgeland. Our aim in this paper is to give a partially new proof of Bridgeland's result using noncommutative rings. The new proof also covers some mild singular and higher-dimensional situations (including those occuring in the recent paper by Chen [11]).

Given a regular function $H\colon\mathbb{R}^{3}\to\mathbb{R}$, we look for $H$-bubbles, that is, regular surfaces in $\mathbb{R}^{3}$ parametrized on the sphere $\mathbb{S}+^{2}$ with mean curvature $H$ at every point. Here we study the case of $H(u)=H_{0}+\varepsilon H_{1}(u)=:H_{\varepsilon}(u)$, where $H_{0}$ is a nonzero constant, $\varepsilon$ is the smallness parameter, and $H_{1}$ is any $C^{2}$-function. We prove that if $\bar p\in\mathbb{R}^{3}$ is a ``good'' stationary point for the Melnikov-type function $\Gamma(p)=-\int_{|q-p|<|H_{0}|^{-1}}H_{1}(q)\,dq$, then for $|\varepsilon|$ small there exists an $H_{\varepsilon}$-bubble $\omega^{\varepsilon}$ that converges to a sphere of radius $|H_{0}|^{-1}$ centered at $\bar p$, as $\varepsilon\to 0$.

We study the space of $L^2$ harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type. These metrics generalize the geometric structures at infinity of several different well-known classes of metrics, including asymptotically locally Euclidean manifolds, the (known types of) gravitational instantons, and also Poincaré metrics on $\mathbb{Q}$-rank $1$ ends of locally symmetric spaces and on the complements of smooth divisors in Kähler manifolds. The answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold. The $L^2$ signature formula implied by our result is closely related to the one proved by Dai [25] and more generally by Vaillant [67], and identifies Dai's $\tau$-invariant directly in terms of intersection cohomology of differing perversities. This work is also closely related to a recent paper of Carron [12] and the forthcoming paper of Cheeger and Dai [17]. We apply our results to a number of examples, gravitational instantons among them, arising in predictions about $L^2$ harmonic forms in duality theories in string theory.

We study distribution of orbits of a lattice $\Gamma\subseteq\SL(n,\mathbb{R})$ in the the space $\mathcal{V}_{n,l}$ of $l$-frames in $\mathbb{R}^n$ ($1\le l\le n-1$). Examples of dense $\Gamma$-orbits are known from the work of Dani, Raghavan, and Veech. We show that dense orbits of $\Gamma$ are uniformly distributed in $\mathcal{V}_{n,l}$ with respect to an explicitly described measure. We also establish an analogous result for lattices in ${\rm Sp}(n,\mathbb{R})$ which act on the space of isotropic $n$-frames.

All the results in this paper are conditional on the Riemann hypothesis for the $L$-functions of elliptic curves. Under this assumption, we show that the average analytic rank of all elliptic curves over $\mathbb{Q}$ is at most 2, thereby improving a result of Brumer [2]. We also show that the average within any family of quadratic twists is at most $3/2$, improving a result of Goldfeld [3]. A third result concerns the density of curves with analytic rank at least $R$ and shows that the proportion of such curves decreases faster than exponentially as $R$ grows. The proofs depend on an analogue of Weil's ``explicit formula.''